MT1100 Problem Sheet 8 Due: Thursday 26th November 2020 at 10:00. Please write your name clearly at the top of each page. Scan your work as a single-file PDF. Do not use too high a resolution to ensure the file size remains below 20 MB (the upper limit is 50MB). Then upload your PDF file onto Moodle, as per the instructions available from the General Information box on Moodle. You are encouraged to leave your handwritten comments to the marker on the first or last page of your homework submission. 1. Let a > 0 be a constant and let f : R -> R be defined by f(x) = x3 + ax. (i) What are the fixed points of f? Justify your answer. (ii) Compute the stability of the fixed points when a ¥ 1. (iii) A fixed point u is superstable when f'(u) = 0. For which values of a does f have superstable fixed points? 2. Let f : R -> R be given by f(x) = 2 - (x-1)2. (i) What are the fixed points of f? Justify your answer. (ii) Write down an explicit formula for f(2)(x). Find the fixed points of f(2) (computing f(2)(1) might help!), and hence find the limit cycles of period 2 for f. (iii) Which of the above fixed points and limit cycles are stable? Justify your answers.